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Ch. 4 - Exponential and Logarithmic Functions
Blitzer - College Algebra 8th Edition
Blitzer8th EditionCollege AlgebraISBN: 9780136970514Not the one you use?Change textbook
All textbooksBlitzer 8th EditionCh. 4 - Exponential and Logarithmic FunctionsProblem 33
Chapter 5, Problem 33

Use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. logb(xy3z3)\(\log\)_{b}\(\left\)(\(\frac{\sqrt{x}\)y^3}{z^3}\(\right\))

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Start with the given logarithmic expression: \(\log_{b} \left( \frac{\sqrt{x} y^{3}}{z^{3}} \right)\).
Use the logarithm property for division: \(\log_{b} \left( \frac{M}{N} \right) = \log_{b} M - \log_{b} N\). So rewrite the expression as \(\log_{b} (\sqrt{x} y^{3}) - \log_{b} (z^{3})\).
Apply the logarithm property for multiplication: \(\log_{b} (MN) = \log_{b} M + \log_{b} N\). So expand \(\log_{b} (\sqrt{x} y^{3})\) as \(\log_{b} (\sqrt{x}) + \log_{b} (y^{3})\).
Rewrite the radicals and exponents inside the logarithms using the power rule: \(\log_{b} (x^{1/2}) + \log_{b} (y^{3}) - \log_{b} (z^{3})\).
Use the power rule of logarithms: \(\log_{b} (a^{c}) = c \log_{b} (a)\), to write the expression as \(\frac{1}{2} \log_{b} (x) + 3 \log_{b} (y) - 3 \log_{b} (z)\).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Properties of Logarithms

Properties of logarithms include the product, quotient, and power rules, which allow the expansion or simplification of logarithmic expressions. For example, log_b(MN) = log_b(M) + log_b(N), log_b(M/N) = log_b(M) - log_b(N), and log_b(M^k) = k·log_b(M). These rules help break down complex expressions into simpler parts.
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Radicals and Exponents in Logarithms

Radicals can be rewritten as fractional exponents, such as √x = x^(1/2). This conversion allows the use of the power rule of logarithms to simplify expressions involving roots. Recognizing and converting radicals is essential for expanding logarithmic expressions fully.
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Simplifying Logarithmic Expressions Without a Calculator

When possible, logarithmic expressions should be simplified using algebraic manipulation rather than numerical approximation. This involves applying logarithm properties to rewrite expressions in terms of simpler logs or constants, enabling exact answers and deeper understanding of the expression's structure.
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Related Practice
Textbook Question

Begin by graphing f(x) = 2x. Then use transformations of this graph to graph the given function. Be sure to graph and give equations of the asymptotes. Use the graphs to determine each function's domain and range. If applicable, use a graphing utility to confirm your hand-drawn graphs. g(x) = 2.2x

Textbook Question

In Exercises 32–35, the graph of a logarithmic function is given. Select the function for each graph from the following options: f(x) = log x, g(x) = log(-x), h(x) = log(2-x), r(x)= 1+log(2-x)

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Textbook Question

Evaluate each expression without using a calculator. log2 (1/√2)

Textbook Question

Solve each exponential equation in Exercises 23–48. Express the solution set in terms of natural logarithms or common logarithms. Then use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. e(1−5x)=793

Textbook Question

Evaluate each expression without using a calculator. log64 8

Textbook Question

Solve each exponential equation in Exercises 23–48. Express the solution set in terms of natural logarithms or common logarithms. Then use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. 3e5x=1977

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